PROBABILITY RULES

Suppose that a random process results in a number of different outcomes. A sample space is a list of all such outcomes. As an example, suppose I'm interested in the amount of time (in minutes) it takes to drive to work this morning. Based on my past experience, I know that there are four different possibilities. So my sample space looks like the following:

OUTCOME
it takes under 30 minutes
it takes between 30 and 35 minutes
it takes between 35 and 40 minutes
it takes over 40 minutes

I wish to assign probabilities to these four outcomes. Before we actually attach numbers to these outcomes, we should first ask: Are there any rules that probabilities must satisfy?

Yes, probabilities must satisfy three general rules:

• Any probability assigned must be a nonnegative number.

• The probability of the sample space (the collection of all possible outcomes) is equal to 1.

• If you have two outcomes that can't happen at the same time, then the probability that either outcome occurs is the sum of the probabilities of the individual outcomes.

How do we use these rules to assign probabilities in the above "drive to work" example? The first rule tells us that probabilities can't be negative, so it makes no sense to assign -1, say, to the outcome "it takes over 30 minutes". The second and third rules tell us that the probabilities that we assign to a collection of nonoverlapping outcomes must add to 1. Nonoverlapping outcomes means that they can't occur at the same time. For example, the outcomes "takes over 20 minutes" and "takes under 30 minutes" are overlapping since they both can happen (if, say, the trip takes 24 minutes). The outcomes "takes under 20 minutes" and "takes over 25 minutes" are nonoverlapping, since at most one of these events can happen at the same time.

With these rules in mind, here are three hypothetical assignments of probabilities, corresponding to four people, Max, Joe, Sue, and Mary.

FOUR SETS OF PROBABILITIES
OUTCOME Max Joe Sue Mary
it takes under 30 minutes .3 .2 .4 0
it takes between 30 and 35 minutes -.1 .3 .4 .2
it takes between 35 and 40 minutes .4 .4 .1 .8
it takes over 40 minutes .4 .3 .1 0

Who has made legitimate probability assignments in the above table? There are problems with the probabilities that Max and Joe have assigned. Joe can't give the outcome "it takes between 30 and 35 minutes" a negative probability, no matter how unlikely this outcome. Joe has made a mistake, since the sum of his probabilities for the four nonoverlapping outcomes is 1.2, which is not equal to 1.

Sue and Mary have give sets of legitimate probabilities, since they are all nonnegative and they sum to 1. But there differences between these two sets of probabilities, which reflect different opinions of these two people about the length of time to work. Sue is relatively optimistic about the time to work, since .8 of her probability is on the outcomes "under 30 minutes" and "between 30 and 35 minutes". In contrast, Mary believes that a trip under 30 minutes will never occur (it has a probability of 0) and it is very probable that it will take between 35 and 40 minutes.

Page Author: Jim Albert (© 1996)
albert@bayes.bgsu.edu
Document: http://www-math.bgsu.edu/~albert/m115/probability/prob_rules.html